A lot of the discussion has been about Science victory games. I find that the optimal policies are different for different victory conditions and sometimes also map dependent. I will share one policy strategy which I like to use for Cultural Victory Games.
Open tradition for the +3

and faster border growth.
Legalism option 1: If you get a
ruin to open in the first 10 turns to open Tradition, it is often quicker not to build the monument and use Legalism for a free monument (also also to get a free monument in your next 3 cities as soon as they are settled.
Full Liberty to help with rapid expansion to 4 cities and faster development with workers. The order is map dependent (do you need a worker or a settler sooner?). The free great person can be a GS to settle near your National College city, a GA to settle near your culture focused city (eventually with Hermitage and Alhambra), or perhaps a GE if there is a wonder which needs to be rushed (perhaps Petra in a good desert location or a key wonder in a low

city). After getting the free settler, buy another and build a 4th to quickly expand to 4 cities.
Legalism option 2: Time legalism after you have finished building monuments and amphitheaters in all 4 of your cities (Petra provides a free amphitheater in 1 city). Then, when you research the Acoustics technology, all 4 cities will get free opera houses and make building Hermitage much faster.
Piety tree should be opened after Liberty. Try to get the bonus

and 10% reduced policy cost as soon as possible. Piety policies can be interlaced with both Tradition and Liberty depending on how fast policies are progressing relative to research.
Finish Tradition to get the extra benefits for your 4 city empire. Free aqueducts, faster growth, and the

and

bonus in the capital.
The remaining two trees can be any mixture depending on strategy. I often take Freedom once the Industrial era opens to get the benefits of specialists, great people, 2x great tile improvements, and bonus

from wonders. If you choose to use Honor, it is a good idea to open honor early (perhaps the second policy after opening Tradition) to get the maximum

from barbarians, but return to finish the tree later (perhaps last after the other 4 trees are full) in the game. Patronage can always be helpful for the bonus

,

, and of course

from city states. If your 4 cities are coastal, Commerce can provide some good benefits and may be a viable option.